Question

Find the exact solution to the exponential equation. (No decimal approximation)

Answer 1

Let's solve the equation:

`\begin{gathered} 54e^(3x+3)=16 \\ e^(3x+3)=(16)/(54) \\ e^(3x+3)=(8)/(27) \\ \ln e^(3x+3)=\ln ((8)/(27)) \\ 3x+3=\ln ((2^3)/(3^3)) \\ 3x+3=\ln ((2)/(3))^3 \\ 3x+3=3\ln ((2)/(3)) \\ 3x=-3+3\ln ((2)/(3)) \\ x=-1+\ln ((2)/(3)) \\ x=-1+\ln 2-\ln 3 \end{gathered}`

Therefore the solution of the equation is:

`x=-1+\ln 2-\ln 3`